The present invention is generally related to hydrocarbon well stimulation, and is more particularly directed to methods for designing matrix treatments. The invention is particularly useful for modeling stimulation treatments, such as designing matrix treatments for subterranean formations penetrated by a wellbore, to enhance hydrocarbon recovery.
Matrix acidizing is a widely used well stimulation technique. The objective in this process is to reduce the resistance to the flow of reservoir fluids due from a naturally tight formation, or even to reduce the resistance to flow of reservoir fluids due to damage. Acid may dissolve the material in the matrix and create flow channels which increase the permeability of the matrix. The efficiency of such a process depends on the type of acid used, injection conditions, structure of the medium, fluid to solid mass transfer, reaction rates, etc. While dissolution increases the permeability, the relative increase in the permeability for a given amount of acid is observed to be a strong function of the injection conditions.
In carbonate reservoirs, depending on the injection conditions, multiple dissolution reaction front patterns may be produced. These patterns are varied, and may include uniform, conical, or even wormhole types. At very low injection rates, acid is spent soon after it contacts the medium resulting in face dissolution. The dissolution patterns are observed to be more uniform at high flow rates. At intermediate flow rates, long conductive channels known as wormholes are formed. These channels penetrate deep into the formation and facilitate the flow of oil. The penetration depth of the acid is restricted to a region very close to the wellbore. On the other hand, at very high injection rates, acid penetrates deep into the formation but the increase in permeability is not large because the acid reacts over a large region leading to uniform dissolution. For successful stimulation of a well it is desired to produce wormholes with optimum density and penetrating deep into the formation.
It is well known that the optimum injection rate to produce wormholes with optimum density and penetration depth into the formation depends on the reaction and diffusion rates of the acid species, concentration of the acid, length of the core sample, temperature, permeability of the medium, etc. The influence of the above factors on the wormhole formation is studied in the experiments. Several theoretical studies have been conducted in the past to obtain an estimate of the optimum injection rate and to understand the phenomena of flow channeling associated with reactive dissolution in porous media. However, existing models describe only a few aspects of the acidizing process and the coupling of the mechanisms of reaction and transport at various scales that play a key role in the estimation of optimum injection rate are not properly accounted for in existing models.
Studies are known where the goal has been to understand wormhole formation and to predict the conditions required for creating wormholes. In those experiments, acid was injected into a core at different injection rates and the volume of acid required to break through the core, also known as breakthrough volume, is measured for each injection rate. A common observation was dissolution creates patterns that are dependent on the injection rate. These dissolution patterns were broadly classified into three types: uniform, wormholing and face dissolution patterns corresponding to high, intermediate and low injection rates, respectively. It has also been observed that wormholes form at an optimum injection rate and because only a selective portion of the core is dissolved the volume required to stimulate the core is minimized. Furthermore, the optimal conditions for wormhole formation were observed to depend on various factors such as acid/mineral reaction kinetics, diffusion rate of the acid species, concentration of acid, temperature, and/or geometry of the system (linear/radial flow).
Network models describing reactive dissolution are known. These models represent the porous medium as a network of tubes interconnected to each other at the nodes. Acid flow inside these tubes is described using Hagen-Poiseuille relationship for laminar flow inside a pipe. The acid reacts at the wall of the tube and dissolution is accounted in terms of increase in the tube radius. Network models are capable of predicting the dissolution patterns and the qualitative features of dissolution like optimum flow rate, observed in the experiments. However, a core scale simulation of the network model requires enormous computational power and incorporating the effects of pore merging and heterogeneities into these models is difficult. The results obtained from network models are also subject to scale up problems.
An intermediate approach to describing reactive dissolution involves the use of averaged or continuum models. Averaged models were used to describe the dissolution of carbonates. Unlike the network models that describe dissolution from the pore scale and the models based on the assumption of existing wormholes, the averaged models describe dissolution at a scale much larger than the pore scale and much smaller than the scale of the core. This intermediate scale is also known as the Darcy scale.
Averaged models circumvent the scale-up problems associated with network models, can predict wormhole initiation, propagation and can be used to study the effects of heterogeneities in the medium on the dissolution process. The results obtained from the averaged models can be extended to the field scale. The success of these models depends on the key inputs such as mass transfer rates, permeability-porosity correlation etc., which depend on the processes that occur at the pore scale. The averaged model written at the Darcy scale requires these inputs from the pore scale. Since the structure of the porous medium evolves with time, a pore level calculation has to be made at each stage to generate inputs for the averaged equation. Averaged equations used in such models describe the transport of the reactant at the Darcy scale with a pseudo-homogeneous model, i.e., they use a single concentration variable. In addition, they assume that the reaction is mass transfer controlled (i.e. the reactant concentration at the solid-fluid interface is zero). However, the models developed thus far describe only a few aspects of the acidization process and the coupling between reaction and transport mechanisms that plays a key role in reactive dissolution is not completely accounted for in these models. Most systems fall in between the mass transfer and kinetically controlled regimes of reaction where the use of a pseudo-homogeneous model (single concentration variable) is not sufficient to capture all the features of the reactive dissolution process qualitatively and that ‘a priori’ assumption that the system is in the mass transfer controlled regime, often made in the literature, may not retain the qualitative features of the problem.
It would therefore be desirable to provide improved averaged models based upon a plurality of scales which describe the influence of different factors affecting acidizing fluid reaction and transport in wormhole formation during matrix stimulation of carbonates, and such need is met, at least in part, by the following invention.